There are many open problems on modular representation theory of finite groups. There is a well-known list of problems of R. Brauer which date to around 1960, about ordinary and modular representations of finite groups. There are other major conjectures ( Broue's Abelian defect group conjecture has already been mentioned in comments). There are Alperin's Weight conjecture, Dade's Projective Conjecture, the Isaacs-Navarro conjecture, and several related conjectures. This is only the tip of the iceberg.

There are many open problems on modular representation theory of finite groups. There is a well-known list of problems of R. Brauer which date to around 1960, about ordinary and modular representations of finite groups. There are other major conjectures ( Broue's Abelian defect group conjecture has already been mentioned in comments). There are Alperin's Weight conjecture, Dade's Projective Conjecture, the Isaacs-Navarro conjecture, and several related conjectures. This is only the tip of the iceberg (and only in one corner of representation theory).